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Transient Radiator Room Heating—Mathematical Model and Solution Algorithm Armin Teskeredzic * and Rejhana Blazevic Faculty of Mechanical Engineering, University of Sarajevo, Sarajevo 71000, Bosnia and Herzegovina; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +387-62-991-918 Received: 2 October 2018; Accepted: 12 November 2018; Published: 18 November 2018

Abstract: A mathematical model and robust numerical solution algorithm for radiator heating of an arbitrary room is presented in this paper. Three separate and coupled transient thermal energy equations are solved. A modified transient heat conduction equation is used for solving the heat transfer at multi-layer outer walls and room assembly. Heat exchange between the inner walls and the observed room are represented with their own transport equation and the transient thermal energy equation is solved for radiators as well. Explicit coupling of equations and linearization of source terms result in a simple, accurate, and stabile solution algorithm. Verification of the developed methodology is demonstrated on three carefully selected test cases for which an analytical solution can be found. The obtained results show that even for the small temperature differences between inner walls and room air, the corresponding heat flux can be larger than the transmission heat flux through outer walls or windows. The benefits of the current approach are stressed, while the plans for the further development and application of the methodology are highlighted at the end. Keywords: radiator heating; mathematical model; numerical solution algorithm

1. Introduction The current energy policy of the EU member states is promoting energy efficiency in buildings as one of the key pillars in the strategic energy documents. The Energy Performance of Buildings Directive [1] imposes very ambitious goals and targets for energy efficient buildings and the topic is strongly promoted by the Energy Efficiency Directive [2]. Programs for ambitious refurbishment of existing buildings and near zero energy standards for new and refurbished buildings are ongoing. The decreased energy need for buildings with different interventions at the building envelope is now introducing new challenges for heating design engineers. From the old-fashioned steady state calculations made only for the nominal conditions and more or less the standard approach of oversizing systems, the philosophy of heating design engineers has to be changed tremendously and shifted towards tight design and optimized control [3]. Engineers are now approaching the field where transient effects for intermittently heated buildings are one of the key issues. Beside the outer insulated walls and their own thermal capacity, the inner walls in massive buildings present large heat reservoirs, either as sources or sinks, and their influence on the building heat balance cannot be neglected. The heat storage effect within the buildings is also recognized as very important from the district heating perspective, where significant daily load variations result in inefficient heat generation. Analysis conducted in [4] is focused on the use of the thermal inertia of buildings in order to reduce daily load peaks and to improve the efficiency of the heating system. In five multifamily residential buildings under investigation, temperature sensor readings were collected for the whole heating season, while different operational schedules were applied in order to minimize the effect of the peak

Buildings 2018, 8, 163; doi:10.3390/buildings8110163

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load. In the other case [5], the influence of thermal inertia in massive buildings is compared to the effects of the hot water tank from the perspective of heat storage. Transient heat conduction in multi-layer walls is usually solved by using either the conduction transfer function [6] or finite difference approach. The finite difference approach is mostly applied in cases where the heat conductivities are temperature-dependent or for the treatment of the phase-change materials within the building’s envelope. For many years, the conduction transfer function for computational reasons was considered as the best choice and is used by default in commercially available energy simulation programs Energy Plus [6] or TRNSYS [7]. These tools are mostly used for analysis of annual energy consumption, where the time step is usually one hour but can be minutes, depending on the simulation scenarios. In contrast, most of the control design processes require time domain simulations in the order of seconds, which makes Energy Plus or TRNSYS unusable for this purpose [8]. The state-of-the-art approach for the dynamic modelling of room heat balance is given in [9] and versions of this model can be also found in [10–16]. The room model, as part of the Modelica Buildings library, covers all physical effects relevant for calculation of a room’s thermal dynamics. It also allows optimization of the heat supply controls with arbitrarily sized time steps. However, more complex physics causes the larger size of the models and limits the applicability of the developed room model in cases of large buildings. In such cases, simplifications are necessary in order to decrease the size of the model. However, in all mentioned examples, the real three-dimensional (3D) heat transfer is converted into a 1D problem and as such, can neither treat the existence of the thermal bridges nor the 3D heat flow within the inner wall structures of the building. This paper introduces a new approach in several aspects: the finite volume method [17] based on a modified transient heat conduction equation is used for the outer walls and room assembly, the discretization process as transformation of the integral-differential transport equations is clearly presented, and implementation of three different solvers is demonstrated. The modelled physics, although not as comprehensive as that mentioned in [9], allows the calculation of the transient room heat balance and can be applied in well-insulated and massive buildings with intermittent heating without any restriction regarding the time step size. One of the key points is the fact that the inner walls’ thermal inertia is decoupled from the room air temperature, allowing the inner walls to be treated as heat sources or sinks, depending on the current temperature gradients between the room air and the inner walls. In parallel with the writing of this paper, an advanced MS Excel tool with visual basic for application (VBA) is developed for validation and testing of the developed procedure. The results of the calculations are validated on three test cases, specifically designed to test all features of the proposed procedure. The overall goal of proposed calculation procedure is to develop a simple tool which will support engineers in the design phase and will in the future be focused on the following core issues:

• • •

Optimization of heat curve parameters for the weather compensated control, Calculation of optimized heat flux for achieving thermal comfort and minimizing energy consumption, Determination of the optimal start time of the heating system after night setback in order to reach the desired temperature without excessive energy consumption.

It is the authors’ vision to expand and adapt the existing Computational Fluid Dynamics software, based on finite volume method (FVM) discretization, and apply it to calculation of the energy need for complex buildings with arbitrary shape of the envelope. This paper describes the core elements of the methodology and calculation procedure in 1D with intention to be expanded in 3D in the future.


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2. Mathematical Model, Numerical Method, and Solution Algorithm 2.1. Modelling Assumptions In this paper, the following assumptions are used to model the room heat balance. A room is presented with one computational control volume, which implies uniform room temperature. The radiator body thermal mass is added to the hot water thermal mass in Equation (3). It is assumed that all inner walls (upper, side, and floor walls) are represented with one thermal mass element. Inner wall conductivity is assumed to be very large, which implies that all surrounding inner walls are at the same temperature and are uniformly heated/cooled. At the opposite side of the inner walls, facing other rooms, an adiabatic boundary condition is assumed. This assumption implies that the surrounding rooms are heated as well as the one under observation. 2.2. Mathematical Model The mathematical model is described with governing Equations (1)–(3) which reflect the behavior of the observed physical system. For multi-layer outer walls and room assembly, a modified transient heat conduction equation is used with additional heat sources and sinks, physically representing the transmission losses through windows, infiltration and ventilation losses, heat exchange with inner walls, and heat gains from the radiators. A separate thermal energy transport equation is solved for inner walls and finally the radiator equation contains transient, convective, and sink terms. The equations are strongly coupled, since the heat flux from radiators and from or to inner walls depend on the room temperature and vice versa. In the text which follows, the mathematical model with modelling assumptions is presented. ∂ ∂t

Z

ρcTdV =

V

I

k∇ T ·ds + h− Qwin − Qiwa − Qven + Qs,i + Qrad i

(1)

S

∂ ∂t ∂ ∂t

Z Viwa

Z V

ρiwa ciwa Tiwa dV =

Z

k iwa ∇ Tiwa · ds + Qiwa

(2)

S∗

(ρw cw + ρrb crb ) Tm dV +

I

ρw cw Tw v·ds = − Qrad

(3)

S

where t is the time, ρ is the density, c is the specific heat, T is temperature, k ∇ T is the heat flux vector due to conduction heat transfer defined by Equation (4), ds is the surface vector perpendicular to the conductive heat flux vector, Qwin are the windows transmission losses, Qiwa is the heat exchange between the room and inner walls, Qven is ventilation/infiltration heat sink, Qs,i is the heat source from solar or/and internal gains, and Qrad is the heat flux exchanged between the radiator and observed room. Subscript iwa in Equation (2) denotes internal wall values, subscript w denotes water and subscript rb denote radiator body physical properties. Note that the terms in < > brackets in Equation (1) are valid for room only and they disappear in the part of the multi-layer walls where the pure transient heat conduction equation is solved. The constitutive relationship between the heat flux and the temperature gradient for isotropic materials is given by Fourier’s law, Equation (4), and is already incorporated into transport Equation (1) within the first term at the right-hand side, as: q = −k∇ T

(4)

where q is the conductive heat flux, k is the thermal conductivity, and ∇ T is the temperature gradient. Equation (2) describes the influence of the thermal capacity of the inner walls on heated rooms. These inner walls are at a different temperature from the room air and their large thermal capacity


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plays the role of a heat reservoir, source or sink, depending on the temperature gradient between the roomSince and the internal walls. This is 𝑘especially valid for intermittently heated buildings. inner wall conductivity 𝑖𝑤𝑎 is assumed to be very large, the first term on the right-hand inner wall conductivity is assumed very large, the first term on the right-hand side Since of Equation (2) disappears, duek iwa to the fact that to ∇𝑇be 𝑖𝑤𝑎 = 0. Equation (2) is coupled to Equation (1) side of Equation (2) disappears, due to the fact that ∇ T = 0.depends Equationon (2)the is coupled to Equation (1) iwa flux via the convective heat transfer term 𝑄𝑖𝑤𝑎 and this heat temperature difference via the convective heat transfer term Q and this heat flux depends on the temperature difference iwa between the inner wall and room temperature. between the inner wall and room temperature. Equation (3) describes behavior of the radiators and 𝑇𝑚 represents the lumped temperature of Equation 𝑇(3) describes behavior of the radiators and Tm represents the lumped the radiator, inlet and outlet of the temperature radiator andof𝑄the 𝑤 is the temperature of the flowing water at 𝑟𝑎𝑑 radiator, Tw sink is theortemperature of from the flowing water at inlet outlet the radiator the rad isfrom is the heat the heat flux radiator towards theand room air.ofNote that the and heatQflux heat sink or heat flux from radiator towards the walls room and air. Note the heat from radiators radiators is the a heat source for the multi-layer outer roomthat assembly in flux Equation (1), whileisit aisheat source for the multi-layer outer walls and room assembly in Equation (1), while it is the heat the heat sink in the radiator Equation (3). This means that Equations (1) and (3) are coupled via the sink radiator Equation (3). This means that Equations (1) and (3) are coupled via the term Q . rad termin𝑄the . 𝑟𝑎𝑑 2.3. Numerical Method 2.3. Numerical Method This paper proposes different approaches for solving the three coupled transport equations as This paper proposes different approaches for solving the three coupled transport equations as proposed in the mathematical model section of the paper. Figure 1a represents the physical domain of proposed in the mathematical model section of the paper. Figure 1a represents the physical domain the model room, while Figure 1b denotes the computational domain. of the model room, while Figure 1b denotes the computational domain.

(a)

(b)

Figure1.1.(a) (a)Physical Physicaldomain domainof ofthe themodel modelroom; room;(b) (b)Calculation Calculationdomain. domain Figure

Variables Variables in in the the rectangular rectangular boxes boxes in in Figure Figure 1a 1a are are known known values, values, such such as as T𝑇out as ambient ambient 𝑜𝑢𝑡 as . temperature, temperature, T𝑇S𝑆 as the radiator supply temperature, mass flow in the radiators m 𝑚̇ is is given given in in aa triangle, triangle, since since its its value value has has to to be be known known and and can can be be either either calculated calculated from from the the nominal nominal conditions conditions or or measuredvalues values(see (see Section the variables presented insuch circles, suchas as 𝑇𝑟𝑜𝑜𝑚 as room measured Section 3.3), 3.3), the variables presented in circles, as Troom room temperature, Q as radiator heat flux, T as average internal wall temperature, and T as the return temperature temperature, 𝑄 as radiator heat flux, 𝑇 as average internal wall temperature, and 𝑇 R iwa 𝑟𝑎𝑑 𝑖𝑤𝑎 𝑅 as the rad return temperature unknown. At the Figure 1b, is thepresented calculation domain is presented are unknown. At theare Figure 1b, the calculation domain with graphical symbols ofwith the multi-layer wall and assembly wall as domain 1 with solver 1as(S1), inner1 walls as domain with graphical symbols of room the multi-layer and room assembly domain with solver 1 (S1),2 inner walls as domain with solveras 2 (S2), and3the radiator domain 3 with solver 3 (S3). solver 2 (S2), and 2the radiator domain with solver as 3 (S3). Figure Figure 22 is isthe the graphical graphical representation representation of ofthe thecoupling coupling procedure procedurein intime. time. An An explicit explicitcoupling coupling procedure is is adopted, adopted, which which means means that that the the room room temperature temperature is is calculated calculated based based on on the the previous previous procedure timestep stepvalues values of room the room temperature to transient term discretization), inner wall time of the temperature (due to (due transient term discretization), inner wall temperature, temperature, and return temperature from the radiator,contained implicitly in contained the heat and return temperature from the radiator, implicitly the heatin flux termflux Qradterm in 𝑄𝑟𝑎𝑑 in Equations (1) In and (3). follows, In what the follows, the discretization detailed discretization of systems differentwill systems will be Equations (1) and (3). what detailed of different be outlined. outlined.

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Figure 2. Graphical of the the explicit explicit coupling coupling procedure. procedure. Figure 2. Graphical representation representation of

For every transport equation the calculation details, boundary, and initial conditions are given, and at the end the overall solution algorithm algorithm is is proposed. proposed. 2.3.1. Discretization of Multi-Layer Outer Walls Walls and and Room Room Assembly Assembly For the multiple-layer outer wall and room assembly, the finite volume method (FVM) discretization principles are used. Discretization of of space, space, time, and equations is implemented used. Discretization implemented according to [17,18]. The space domain is divided into the finite number number of control volumes (CVs), (CVs), not necessarily of the same size, where the calculation points are located at the cell centers (collocated necessarily of the same size, where the calculation points are located at the cell centers variable arrangement). (collocated variable arrangement). The Euler’s scheme of implicit integration in time is used, where the calculation time is divided into the not necessarily of the same duration. ThisThis scheme is ofisthe the final finalnumber numberofoftime timeintervals, intervals, not necessarily of the same duration. scheme offirst the order of accuracy but is unconditionally stabile, irrespective of the time of step size. Spatial first order of accuracy but is unconditionally stabile, irrespective the time step discretization size. Spatial is performed assuming a linear variation of the dependent between variable adjacent between calculation points discretization is performed assuming a linear variation ofvariable the dependent adjacent (cell centers)points and the integrals by using the mid-point rule, ensuring therule, second order calculation (cell centers)are andcalculated the integrals are calculated by using the mid-point ensuring accuracy. Diffusion are calculated at the cell-face centers asatthethe weighting the second order coefficients accuracy. Diffusion coefficients are calculated cell-faceaverage centersbetween as the cell center values. weighting average between cell center values. By applying applying adopted adopted discretization discretizationprinciples, principles,the theresulting resultingheat heat sources, Equations (5)–(8), sources, Equations (5)–(8), in in the thermal energy Equation (1) for the multi-layer wall and room assembly are calculated as: the thermal energy Equation (1) for the multi-layer wall and room assembly are calculated as: Z

Q𝑄win = U 𝑤𝑖𝑛((𝑇 Troom − T 𝑜𝑢𝑡))𝑑𝑆 dS ≈ ( Troom − ) S 𝑤𝑖𝑛 ≈U 𝑈win −T 𝑇out 𝑤𝑖𝑛 = ∫ 𝑈win 𝑟𝑜𝑜𝑚 − 𝑇out 𝑤𝑖𝑛 (𝑇𝑟𝑜𝑜𝑚 𝑜𝑢𝑡 ) 𝑆win

(5) (5)

S𝑆win 𝑤𝑖𝑛

Z

Qiwa = αiwa ( Troom − Tiwa ) dS ≈ αiwa ( Troom − Tiwa ) Siwa 𝑄𝑖𝑤𝑎 = ∫ 𝛼𝑖𝑤𝑎 (𝑇𝑟𝑜𝑜𝑚 − 𝑇𝑖𝑤𝑎 ) 𝑑𝑆 ≈ 𝛼𝑖𝑤𝑎 (𝑇𝑟𝑜𝑜𝑚 − 𝑇𝑖𝑤𝑎 ) 𝑆𝑖𝑤𝑎

(6) (6)

Siwa

𝑆𝑖𝑤𝑎

nh ρc( Troom − Tout ) Qven = dV ≈ 𝑛ℎ 𝜌𝑐(𝑇3600 𝑟𝑜𝑜𝑚 − 𝑇𝑜𝑢𝑡 ) 𝑄𝑣𝑒𝑛 = Vroom ∫ 𝑑𝑉 ≈ 3600 Z

𝑉𝑟𝑜𝑜𝑚

nh 𝑛ℎ ρc( Troom − Tout ) Vroom 3600 𝜌𝑐(𝑇𝑟𝑜𝑜𝑚 − 𝑇𝑜𝑢𝑡 ) 𝑉𝑟𝑜𝑜𝑚 3600 Z ∆Tm n N Qrad = qrad dA ≈ Qrad N ∆T ∆𝑇m𝑚 𝑛 𝑁 𝑄𝑟𝑎𝑑 =A ∫ 𝑞𝑟𝑎𝑑 𝑑𝐴 ≈ 𝑄𝑟𝑎𝑑 ( 𝑁) ∆𝑇𝑚 𝐴

(7) (7) (8) (8)

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where 𝑈𝑤𝑖𝑛 is the heat transfer coefficient for windows, 𝑇𝑜𝑢𝑡 is the ambient temperature, 𝑆𝑤𝑖𝑛 is thewhere area of windows, 𝛼𝑖𝑤𝑎transfer is the heat transfer coefficient theambient inner walls and room,S𝑆win Uwin is the heat coefficient for windows,between Tout is the temperature, the 𝑖𝑤𝑎isis thearea areaofofwindows, inner walls in is contact with the room, 𝑛ℎ is the number air changes per room, hour, S 𝐴iwa is is thethe αiwa the heat transfer coefficient between theofinner walls and 𝑁 surface the walls radiator body, 𝑄 is the nominal heat flux of thechanges radiatorper defined nominal area ofofinner in contact with room, nh is the number of air hour, Afor is the surface 𝑟𝑎𝑑 the 𝑁N conditions, ∆𝑇𝑚 and ∆𝑇Q areismean temperature difference radiator and the room for of the radiator body, the nominal heat flux of the between radiator the defined for nominal conditions, 𝑚 rad N are mean arbitrary and conditions, respectively, and 𝑛 is the radiator exponent. Note that source ∆Tm and ∆Tnominal temperature difference between the radiator and the room for arbitrary and m terms given by Equations (5)–(8) are valid for the room only and they vanish in the multi-layer wall. nominal conditions, respectively, and n is the radiator exponent. Note that source terms given by The heat source dueare tovalid solarfor and gains 𝑄𝑠,𝑖 is vanish known in and in advance as The a function of Equations (5)–(8) theinternal room only and they theset multi-layer wall. heat source time. due to solar and internal gains Qs,i is known and set in advance as a function of time. Enumeration ofof control layers towards towardsthe theroom room(see (seeFigure Figure1b), Enumeration controlvolumes volumesisisdone donefrom from the the outside outside layers 1b), where first is contact in contact with environment represents room where thethe first CVCV is in with thethe environment andand thethe lastlast CVCV represents the the room withwith index index N + 1, where N is the sum of all CVs defined in the solid multi-layer wall. Note that number N + 1, where N is the sum of all CVs defined in the solid multi-layer wall. Note that number of of CVs CVs arbitrarily each layer of the wall. cancan be be arbitrarily setset forfor each layer of the wall. In In order toto demonstrate Equation (1) (1) will willbe order demonstratethe thediscretization discretizationpractices, practices,the thebalance balance according according to to Equation bewritten writtenin indiscrete discreteform formfor forthree threeneighboring neighboring CVs, CVs, given given by by Equations Equations (9) (9) and and (10) (10) and and presented presented in in Figure Figure 3. 3.

Figure 3. Three neighboring CVs with compass notation (P—central cellcell center, W—west cellcell center, Figure 3. Three neighboring CVs with compass notation (P—central center, W—west center, E—east cell center). E—east cell center).

Figure 3 shows three neighboring cells with introduced compass notation where observed Figure 3 shows three neighboring cells with introduced compass notation where thethe observed cell, which balance will given, is cell while left-hand side west is marked with cell, forfor which thethe balance will bebe given, is cell P, P, while thethe left-hand side or or west cellcell is marked with letter and right-hand side east cell is marked with letter letter WW and thethe right-hand side or or east cell is marked with letter E. E. The discretized form of of thethe transient term in in Equation (1)(1) becomes: The discretized form transient term Equation becomes: Z i 𝜕∂ 𝛿𝑉δV h m −1 𝑚−1 ρcTdV≈ ≈ 𝑃 [ P(𝜌𝑐𝑇) − ρcT ( ) (ρcT ) ] (𝜌𝑐𝑇) ∫ 𝜌𝑐𝑇𝑑𝑉 − 𝑃 P 𝑃 P 𝜕𝑡∂t 𝛿𝑡 δt

(9) (9)

𝑉V

where 𝛿𝑉δV isisthe CV with withcenter centerP,P,δt𝛿𝑡 is the time subscript P denotes 𝑃 P where thevolume volume of of the the CV is the time stepstep size,size, subscript P denotes values values from the cell center P, while the exponent 𝑚 − 1 denotes values from the previous time step. from the cell center P, while the exponent m − 1 denotes values from the previous time step. The discretized form of of thethe diffusion flux in in Equation (1)(1) becomes: The discretized form diffusion flux Equation becomes: I 𝑘𝑒k e 𝑘𝑤k w (𝑇𝑃( T−P 𝑇 ∮ 𝑘∇𝑇 ∙ d𝐬 𝑇𝑃T)P𝑆)𝑒S− k∇T ·ds≈≈𝛿 (𝑇 −𝑊T)W𝑆𝑤) Sw (T 𝐸 E−− e− 𝛿 δ𝑃𝐸PE δWP 𝑊𝑃

(10)(10)

𝑆S

where 𝑘𝑒 and 𝑘𝑤 are the conductivities calculated at the cell face surfaces e (between cell E and P) where k e and k w are the conductivities calculated at the cell face surfaces e (between cell E and P) and w (between cells W and P), respectively, 𝛿𝑃𝐸 and 𝛿𝑊𝑃 are distances between cell centers P and and w (between cells W and P), respectively, δPE and δWP are distances between cell centers P and E and W and P, respectively, and 𝑇𝑃 , 𝑇𝑊 , and 𝑇𝐸 are values of the temperatures in cell centers P, W, E and W and P, respectively, and TP , TW , and TE are values of the temperatures in cell centers P, W, and E, respectively. Note that the first term on the right-hand side of Equation (10) is the heat flux at and E, respectively. Note that the first term on the right-hand side of Equation (10) is the heat flux at the east cell face surface, while the second term represents the heat flux at the west cell face surface.


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the east cell face surface, while the second term represents the heat flux at the west cell face surface. The cell face value of conductivity, Equation (11), for example at cell face e, is according to [18] and is The cell face value of conductivity, Equation (11), for example at cell face e, is according to [18] and is calculated as: calculated as: 𝑘k𝑒e 11 = δ (11) = Pe eE (11) δ𝛿PE 𝛿 +𝛿δk𝑒𝐸 𝑃𝑒 𝑃𝐸 kP + E 𝑘𝑃 𝑘𝐸 where δPe is the distance between cell center P and cell face e, δeE is the distance between cell face where 𝛿𝑃𝑒 is the distance between cell center P and cell face e, 𝛿𝑒𝐸 is the distance between cell face e e and cell center E (see Figure 3), while k P and k E are the conductivities in cell centers P and E, and cell center E (see Figure 3), while 𝑘𝑃 and 𝑘𝐸 are the conductivities in cell centers P and E, respectively. Note that usual approach is to find the linear interpolation of conductivities at the cell respectively. Note that usual approach is to find the linear interpolation of conductivities at the cell faces, which, in the case of large differences of conductivities in CVs P and E, introduces calculation faces, which, in the case of large differences of conductivities in CVs P and E, introduces calculation error. By using the current approach, the diffusive coefficient at the cell face is calculated correctly. error. By using the current approach, the diffusive coefficient at the cell face is calculated correctly. A special treatment is necessary for the first and last cell in the multi-layer wall, which is given by A special treatment is necessary for the first and last cell in the multi-layer wall, which is given Equations (12) and (13). If point P is the first cell which neighbors the ambient environment (Figure 4 by Equations (12) and (13). If point P is the first cell which neighbors the ambient environment left side), then the west cell face conductivity is calculated as: (Figure 4 left side), then the west cell face conductivity is calculated as: 1 w1 𝑘k𝑤1 = 1 1 δ (12) = B1 δ B1 (12) + 𝛿k𝐵1 𝛿𝐵1 α1out + 1 𝛼𝑜𝑢𝑡 𝑘1 where αout is the heat transfer coefficient from the outside wall to environment. In the case that point P where 𝛼𝑜𝑢𝑡 is the heat transfer coefficient from the outside wall to environment. In the case that is neighboring the room (Figure 4 right), the cell face conductivity is calculated according to: point P is neighboring the room (Figure 4 right), the cell face conductivity is calculated according to: 𝑘k e𝑒 11 = δ = 1 Ne δ𝛿Ne 𝛿 𝑁𝑒 + 1 𝑁𝑒 k + αin 𝑘𝑁N 𝛼𝑖𝑛

(13) (13)

where α𝛼in is the heat transfer coefficient from the room to the outside-facing wall. where 𝑖𝑛 is the heat transfer coefficient from the room to the outside-facing wall.

Figure Figure 4. 4. Special Specialcalculation calculationofofcoefficients coefficients and and source source terms terms for for marked marked surfaces surfaces (exposed (exposed to to ambient—left hand side and at interface outer wall and room—right hand side). ambient—left hand side and at interface outer wall and room—right hand side).

The The final final form form of of the the discretized discretized Equation (14) is written in the form: 𝑎a𝑃 𝑇 −∑𝑎 𝑇 = 𝑏 P T𝑃P − ∑ a j𝑗T𝑗j = b𝑃 P

(14) (14)

𝑗j

where index j = {E,W} denote position of the CV according to compass notation, 𝑎𝑃 is the central where index j = {E,W} denote position of the CV according to compass notation, a P is the central coefficient with values near 𝑇𝑃 , 𝑎𝑗 are the neighbour coefficients new temperature values 𝑇𝑗 , and coefficient with values near TP , a j are the neighbour coefficients new temperature values Tj , and the the right hand side is the source term 𝑏𝑃 . right hand side is the source term bP . For exemplary cell P given in Figure 3, the values of the neighboring and central coefficients as For exemplary cell P given in Figure 3, the values of the neighboring and central coefficients as well as the source terms are given in Equations (15)–(17) as: well as the source terms are given in Equations (15)–(17) as: 𝑆𝑒 𝑆𝑤 𝑎𝐸 = 𝑎S𝑊w = Se (15) 𝛿𝑃𝑒 a 𝛿= 𝛿 𝛿 𝑒𝐸 𝑤𝑊 aW = δ (15) +E + 𝑤𝑃 wW 𝑘𝑃 𝑘𝐸 δPe + δeE 𝑘𝑃 + δwP𝑘𝑊 kP

𝑎𝑃 =

kE

kW

kP

𝛿𝑉𝑃 𝜌𝑃 𝑐𝑃 𝑛ℎ + 𝑎𝐸 + 𝑎𝑊 +< 𝑈𝑤𝑖𝑛 𝑆𝑤𝑖𝑛 + 𝛼𝑖𝑤𝑎 𝑆𝑖𝑤𝑎 + 𝛿𝑉 𝜌 𝑐 > 𝛿𝑡 3600 𝑃 𝑃 𝑃

(16)


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aP = bP =

δVP ρ P c P n + a E + aW + hUwin Swin + αiwa Siwa + h δVP ρ P c P i δt 3600

(16)

n δVP ρ P c P m−1 ∗ TP + hUwin Swin Tout + αiwa Siwa Tiwa + h δVP ρ P c P Tout + Qs,i + Qrad i δt 3600

(17)

If for all CVs within the solution domain of multi-layer walls and room assembly one writes coefficient and source terms, now by following enumeration given in Figure 1b, a system of N + 1 linear equations will be obtained. Note that the part of the heat flux in expressions (5)–(7), containing the room temperature are transferred to the central coefficient in Equation (16). The remaining parts of these fluxes are part of the source terms in Equation (17) and contain known values such as ambient temperatures Tout and inner wall temperature Tiwa (explicit coupling). The system of equations can also be written in the vector format, given by Equation (18) as: Ax=b

(18)

where A is the coefficient matrix, x is the vector of the dependent variable—in our case temperature— and b is the vector of the source terms. By preserving introduced compass notation, Equation (18) can be also written in a matrix form, Equation (19), as:         

a P1 aW2 0 ··· 0 0

a E1 a P2 aW3 ··· 0 0

0 a E2 a P3 ··· 0 0



0 0 a E3 ···

0 0 0 ···

0 0 0 ···

aW ( N ) 0

a P( N ) aW ( N + 1 )

a E( N ) a P ( N +1)

    ·   

T1 T2 T3 ··· TN TN +1





        =      

bP1 bP2 bP3 ··· bP( N ) b P ( N +1)

        

(19)

where a Pi are the central coefficients or previously declared as a P , for example, given in Figure 3 and Equation (16), where i is the counter of CVs and goes from I = 1 to N + 1 CVs, while N + 1 represents the index of the room. Note that the matrix is tridiagonal and all source terms bPi are known, since the ∗ featuring in explicit coupling procedure is adopted. Note that the heat flux from the radiators Qrad Equation (17) is calculated based on the Troom value taken from the previous time step. It means that the system given in Equation (19) can be solved by a direct algorithm known as the three diagonal matrix algorithm (TDMA) or the Thomas algorithm. All temperature values are obtained in two passes. First the system, Equation (19) is written in the form of Equation (20) as: aWi Ti−1 + a Pi Ti + a Ei Ti+1 = bi ,

(i = 1, 2, . . . , N + 1)

(20)

TDMA uses Gaussian elimination to put the system in upper triangular form by computing the new diagonal coefficients a Pi and the new right-hand side vector components bi given by Equation (21), while the unknown temperatures are obtained from the backward substitution as given by Equation (22): a Pi = a Pi −

aWi a P ( i −1)

a E ( i −1) ,

bi = bi −

aWi a P ( i −1)

b( i − 1 ) ,

(i = 2, 3, . . . , N + 1)

(21)

where the equal sign means that the value is replaced by. TN +1 =

b N +1 a P ( N +1)

,

Ti =

bi − a Ei Ti+1 , a Pi

(i = N, N − 1, . . . , 1)

(22)

The original set of integral-differential equations is transformed into algebraic equations by the discretization practices described above. Linearization of the source terms is implemented by implicit treatment of heat sources/sinks as part of the coefficient matrix, which together with explicit coupling of equations results in a tridiagonal system of equations solved by TDMA.


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2.3.2. Final Form of the Transport Equation for Inner Walls The assumption of modelling all inner walls with a unit thermal mass element implies the use of one computational cell for inner walls. In this case, the temporal Euler’s explicit scheme is used, while the spatial discretization does not make sense. The algebraic equation which is solved for inner walls, Equation (23), has the final form: old Tiwa = Tiwa +

αiwa Siwa miwa ciwa δt

∗ old Troom − Tiwa

(23)

where miwa is the mass of the inner walls as a product of the inner wall density and volume, superscript ∗ old means that the values are taken from the previous time step and Troom is the last available room temperature, taken from the previous time step. It can be noted that no special solver is needed for the transport equation of the inner walls, since it can be solved explicitly, bearing in mind that all variables at the right-hand side of Equation (23) are already known. 2.3.3. Final Form of Transport Equation for Radiators In order to make the equation valid for radiators, the unknown variables need to be declared for Equation (3). The lumped radiator temperature Tm is in this work calculated as the arithmetic average according to Equation (24): T + TR Tm = S (24) 2 where TS is the radiator water supply temperature and TR is the water return temperature. Note that in the current study, a simple, one-zone heating system is envisaged with weather compensated control, which means that the supply temperature in the radiator is almost the same as the heating system’s supply temperature (no mixing valve in the supply nor the return pipe and negligible thermal losses). Temperature differences ∆Tm and ∆TmN at the right-hand side of Equation (8) are calculated by treating the radiator as a heat exchanger. The consequence is that the log mean temperature difference (LMTD) is used and ∆Tm is calculated by Equation (25): ∆Tm =

TS − TR

(25)

Troom ln TTRs − − Troom

where this expression is valid for arbitrary conditions as well as for the nominal radiator conditions. The governing equation for the radiator (3) is discretized by assuming the mid-point rule for calculation of integrals and by using Euler implicit time integration. In this case, the domain of the radiator body contains only one computational cell and is not discretized into a finite number of calculation points. In the transient calculation of heat transfer from radiators to the surrounding air, the bisection method was used. Equation (26) has been numerically solved, where the U A N value is defined by Equation (27):

(mrb crb + Vw ρw cw ) δt

TS + TR 2

−

Tmold



.

− mcw ( TS − TR ) = −U A

where U AN =

N Qrad N) (∆tm

n

N

ln

n TS − TR  ∗ TS − Troom ∗ TR − Troom

(26)

(27)

N , is the radiator constant which is related to the nominal conditions of the radiator, its power Qrad N and ∆tm the difference between the average radiator surface temperature and the room temperature in nominal conditions and calculated according to (25). Note that constant U A N represents the product


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of the overall heat transfer coefficient and the area of radiator and physically implies the constant U value of the radiator over different temperature regimes and/or mass flows in the radiator. . An additional variable which needs to be determined is mass flux m appearing in Equation (26). Mass flux can be calculated for the nominal temperature regime of the radiator, which is given by producers. However, in most practical situations, the real mass flow in radiator differs from the nominal mass flow. In cases where the measurements of the room temperature exists, where supply and return temperatures are known, the mass flux can be estimated. Equation (26) is solved by the halving interval method (bisection method) iteratively for TR as a dependent variable within an initially prescribed interval: Troom ≤ TR ≤ TS The presented mathematical model and calculation procedure implies known supply temperature, which is always the case when weather compensated control is used. Equation (26) is iteratively solved until the convergence criteria is reached (i.e., less than 1 × 10−4 ). 2.4. Solution Algorithm The solution procedure is executed as follows: 1.

Data for domain of multi-layer walls and room assembly are defined a. b. c.

2.

Data for inner walls are defined a. b.

3.

d.

b.

6.

Nominal radiator capacity, Nominal regime as nominal supply, return, and room temperature, Physical properties of mass and specific heat of the radiator body, including the content of the water in the radiator is prescribed, Relevant mass flow is defined, either taken from nominal conditions or from the measurements (where available),

Initial and boundary conditions are given for all equations; a.

5.

Heat capacity is prescribed with mass of all inner walls and their specific heat, Heat transfer coefficient and surface from inner walls to room air is defined,

Radiator parameters are defined a. b. c.

4.

Dimensions of the room and surface of outer wall are prescribed, Dimensions of windows are defined, Arbitrary number of layers in the multi-layer wall are prescribed with their physical properties and an arbitrary number of control volumes is assigned for each layer,

Initial temperature profile in the outer wall (constant or arbitrary), initial room temperature, initial inner wall temperature, initial hot water supply temperature, and initial lumped temperature of radiators (equal or larger than room temperature) are prescribed, Boundary conditions are applied by assuming known, variable, or fixed outside temperature and known hot water supply temperature (note that in weather compensated heating control both values are known),

The system of Equations (19) is solved by using the TDMA solver and by assuming the known radiator heat flux and the inner wall heat flux from the previous time step, resulting in the new room temperature value, Equation (23) for inner walls is solved, resulting in a new inner wall temperature


7.

8.

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Equation (26) for radiators is solved iteratively to get the new return temperature by using the current values of supply temperature and last available room temperature value from the previous time step, Steps 5 to 7 are repeated until the final time step is reached.

3. Results In this section, the developed methodology is verified and validated on three characteristic test cases. Validation cases are carefully selected in order to test the validity and correctness of all described features. The setups of these cases are explained in detail and the expected results are outlined. The verification process is completed when the results of the calculations correspond to the expected values, which can be obtained analytically. The main features of the selected cases are described below, while the detailed setup and parameters for calculations are outlined in the following subsections:

•

•

•

Case 1 presents the test case in which several features of the tool are tested. Calculations are run transiently with constant outside temperature, leading to the solution of coupled equations to the steady-state result. In this case, the thermal inertia of the inner walls is assumed to be reasonably low. Different initial temperatures are given for the outer wall, room, and inner walls. When the steady-state is reached, the results of calculation are compared to the values of expected room and inner wall temperatures (should be equal) and expected radiator heat flux, which should compensate for heat losses of the room to the environment. Also, the temperature profiles in the multi-layer outer wall are compared against the analytical solutions for four characteristic and different values of outside temperatures, each run as separate calculations. Case 2 demonstrates a case where the thermal inertia of inner walls is huge, which causes the inner wall temperature to remain constant during the calculations, introducing an additional heat loss to the room. Calculation is run transiently and the expected values—prescribed room temperature and constant inner walls’ temperature—are analyzed when the steady-state is reached. The simulation is run for design conditions in terms of outside and set room temperatures. Case 3 is designed to test the interaction between the heat flux from radiators towards the room and the heat flux between the room and the environment. For this purpose, the thermal inertia of the outer, inner walls, and a radiator is neglected, providing zero influence of the transient terms in Equations (1)–(3). Variable outside temperature is prescribed in time and the calculation is run transiently, formally providing the steady-state solutions for every time step due to the mentioned assumptions.

The calculation domain is presented in symbolic form in Figure 5, where all elements used in the calculations are shown. The geometry of the room, composition of multi-layer outside walls, heat transfer coefficients, size of windows, and infiltration rate are identical for cases 1, 2, and 3. Note that the solar and internal gains are neglected (Qs,i = 0) in all test cases since they are strongly transient in nature and because their implementation is straightforward.

due to the mentioned assumptions. The calculation domain is presented in symbolic form in Figure 5, where all elements used in the calculations are shown. The geometry of the room, composition of multi-layer outside walls, heat transfer coefficients, size of windows, and infiltration rate are identical for cases 1, 2, and 3. Note that the solar and internal gains are neglected (𝑄𝑠,𝑖 = 0) in all test cases since they are strongly transient12inof 18 Buildings 2018, 8, 163 nature and because their implementation is straightforward.

Figure Geometryfor fortest testcases cases1,1,2,2,and and3. 3. Figure 5. Geometry

The test room consists of two outer walls with the same structure, each made of 4 layers (materials). There is one window, one radiator, and all inner walls are represented by one block. Table 1 presents the room size and other characteristic values relevant for step 1 in the setup procedure described in the previous section. In the last two columns, the outside and room temperatures are given, which are relevant for calculation of nominal losses of the room and for sizing the installed power of the radiator. Table 1. Geometry and characteristic parameters for test cases 1, 2, and 3. Test Case Cases 1, 2, 3

HT Coeff. W/m2 K

Design Temp. ◦ C

Room Size (B/H/W) in m

Swin

Uwin

nh

Swall

m2

W/m2 K

h−1

m2

αin

αout

tN room

tN out

5/8/3.5

7.05

1.4

0.5

38.45

20

8

20

−16

Two outside walls are defined with 4 layers, as given in Table 2, with different thicknesses and corresponding thermo-physical properties: thermal conductivity, specific heat, and density. Based on the data given in Tables 1 and 2, it is possible to calculate the overall U value of the multi-layer wall and then to calculate the nominal heat losses of the observed room as: n ρcV N N N ∗ Qloss = Uwall Swall + Uwin Swin + h troom − tout + Qiwa (28) 3600 N are the total heat losses in nominal conditions, U where Qloss wall Swall is the product of U value of wall and area of wall representing the coefficient of heat transmission through walls, Uwin Swin is the product of the U value of window and area of window representing the coefficient of transmission losses through the windows, and the last term in brackets represents the coefficient of infiltration-ventilation losses, where nh is the number of air changes per hour, ρc is the product of air density and specific ∗ , is the heat heat, and V is the room volume. The last term on the right-hand side of Equation (28), Qiwa flux between the room and inner wall and will be discussed for all cases.

Table 2. Thermo-physical properties of outer walls. Layer 1 2 3 4

Material Cement mortar EPS Hollow brick Lime mortar

Thickness, m

Thermal Conductivity, W/mK

Specific Heat, J/kgK

Density, kg/m3

0.02

1.40

1000

1800

0.10 0.25 0.02

0.04 0.64 0.81

1450 900 1000

20 1600 1500


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Table 3 contains the relevant properties for the inner wall and radiator data for all test cases. It can be seen that for Case 2, the thermal capacity of the inner wall (product of mass and specific heat) is set as a very large value, while it is zero for Case 3. In contrast, for Case 1, the thermal capacity of the inner wall is defined as a relatively small value. These facts are discussed in more detail in subsections which describe the specific setup for each test case. The last column in Table 3 represents the supply temperature of the heating system, which is, in this paper, taken as the water temperature entering the radiator (no heat losses are envisaged between the boiler and radiator). Table 3. Physical properties of inner walls and radiator data for all test cases. Inner Wall Properties Test Cases miwa kg Case 1 Case 2 Case 3

100 1 × 106 0

ciwa J/kgK Siwa m2 1400 1 × 106 0

Radiator Parameters

αiwa W/m2 K

100

5

0

0

QN rad,

W 90/70/20

Qrad , W 80/60/20

m ˙ kg/s

Supply Temp.

2084.3 2718.1 2084.3

1644.5 2144.5 1644.5

0.019 0.026 0.019

Heat curve

For cases 1 and 2, a constant outside temperature is prescribed, which results in constant water supply temperature based on heat curve values. If one links this assumption to reality, this would mean that the system is working according to weather compensated control, by which the heat flux from the radiators is controlled via water supply temperature. The mass flow for all test cases remains constant during the calculations (values are given in Table 3), by which it is assumed that neither variable speed drive pumps nor thermostatic valves exist in the system. In terms of the boundary conditions, for all test cases, only the outside temperature is prescribed, while the supply water temperature is calculated according to the heat curve parameters. The radiator mass flow is calculated according to the work conditions of the heating system, while all other variables remain unknown, such as the temperatures in the CV centers in multi-layer wall (in all calculation taken as 12 CVs), room and inner wall temperatures (1 + 1 unknown value), the heat flux from the radiator, and water return temperature (1 + 1 unknown value). 3.1. Validation Case 1—Low Thermal Inertia of Inner Walls For this case, a radiator is chosen so that its heat flux in working conditions exactly compensates the total heat losses of the room in nominal conditions (for design temperatures). For the sake of radiator testing, producer data corresponding to a nominal radiator regime 90/70/20 ◦ C is chosen, while it is supposed that the heating system works according to regime of 80/60/20 ◦ C. Note that the first number denotes the water supply temperature, the second number is the water return temperature, and the third number is a room temperature. The mass flow in the radiator is calculated based on the N = 1644.5 W) and according to the working regime nominal room heat loss in design conditions (Qloss of the heating system. For testing purposes, four different outside temperature values are used, run as four different calculations. Based on Equation (1), the value of outside temperature is calculated in order to get the room heat losses as a percentage of nominal room heat losses. The adopted values are 25%, 50%, 75%, and 100% based on nominal room heat losses and the corresponding prescribed outside temperatures are 11 ◦ C, 2 ◦ C, −7 ◦ C, and −16 ◦ C, respectively. Since weather compensated control is used, for every given outside temperature, a corresponding water supply temperature is calculated according to the heat curve parameters. The values are given in the second column of Table 4. Initial conditions for all outside temperatures are the same and are given as: ◦ in ◦ in ◦ tin w (i ) = 19 C ; tiwa = 19 C ; troom = 15 C

(29)


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where tin w (i ) are the initial temperatures in the multi-layer outside wall, where i = 1 to 12 (for 12 CVs), in tin is the initial temeprature of the inner wall, and troom is the initial room temperature. Since the iwa outside temperature is fixed as a constant value, the problem has a unique solution, irrespective of the prescribed initial conditions. The influence of the transient terms in the equations exists at the beginning of the calculations and decreases over time until it completely vanishes when a steady-state is reached. Table 4. Boundary conditions, supply water temperature, and losses. Outside Temperature tout (◦ C)

Water Supply Temp. tS (◦ C)

Heat Losses Qloss (W)

Qloss /Qloss N (%)

−16 −7 2 11

80.0 67.6 54.3 39.7

1644.5 1233.4 822.3 411.1

100 75 50 25

Validation of the model also requires the validation of the heat balance between the room and inner wall. the small thermal capacity of the inner wall is assumed, it is expected Buildings 2018,In 8, this x FORcase, PEERsince REVIEW 14 of 18 that the inner wall temperature will reach the room temperature at the steady-state. assumed that that the the steady-state steady-state condition condition is is reached reached when when the the difference difference between between the the heat heat ItIt isis assumed fluxes, measured measured at at the the exterior exterior and and interior interior surfaces surfaces of of the the multi-layer multi-layer outer wall is less than 0.5%. fluxes, Figure 6a 6a shows shows the the overlapping overlapping room room and and inner inner wall wall temperature temperature profiles profiles in in time time and and the the radiator radiator Figure heat fluxes (Figure four different values of outside temperature. One can confirm that in heat (Figure6b), 6b),both bothforfor four different values of outside temperature. One can confirm ◦ C. In all cases = 𝑡t𝑟𝑜𝑜𝑚 = 20 ℃. In Figure 6b, beside the temporal variation of radiator heat fluxes, the that in all𝑡𝑖𝑤𝑎 cases = t = 20 Figure 6b, beside the temporal variation of radiator heat room iwa steady-state heat flux values are given as data labels. It can be seen that for all four cases, the radiator fluxes, the steady-state heat flux values are given as data labels. It can be seen that for all four cases, heatradiator fluxes are almost as the room heat losses 4). The(Table largest error the heat fluxesthe aresame almost the calculated same as the calculated room(Table heat losses 4).relative The largest betweenerror the calculated radiator flux value and the heat loss value 0.02%, which is relative between the calculated radiator fluxprescribed value androom the prescribed roomisheat loss value practically negligible. This value could beThis further decreased forcing stricter steady-state criteria is 0.02%, which is practically negligible. value could beby further decreased by forcing stricter (less than 0.5% difference in heat at exterior and interior surface). steady-state criteria (less than 0.5%fluxes difference in heat fluxes at exterior and interior surface).

(a)

(b)

Figure6.6.(a) (a)Room Roomtemperature temperaturevalues valuesinintime; time;(b) (b)Radiator Radiatorheat heat flux 𝑄𝑟𝑎𝑑 , for four different values Figure flux Qrad , for four different values of of outside temperature. outside temperature.

Figure Figure 77 shows shows the the comparison comparison between between analytical analytical results results and and the the results results of of calculation calculation for for four four different It It can be be seen thatthat perfect matching is achieved, proving that differentvalues valuesofofoutside outsidetemperature. temperature. can seen perfect matching is achieved, proving all the features passed the verification process. that alltested the tested features passed the verification process.

Figure 6. (a) Room temperature values in time; (b) Radiator heat flux 𝑄𝑟𝑎𝑑 , for four different values of outside temperature.

Figure 7 shows the comparison between analytical results and the results of calculation for four different Buildings 2018,values 8, 163 of outside temperature. It can be seen that perfect matching is achieved, proving 15 of 18 that all the tested features passed the verification process.

Figure 7. Comparison of the numerical calculation results withwith the analytical results for different values Figure 7. Comparison of the numerical calculation results the analytical results for different values oftemperature. ambient temperature. of ambient Buildings 2018, 8, x FOR PEER REVIEW

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3.2.3.2. Validation Case Validation Case2—High 2—HighThermal ThermalInertia Inertiaof of Inner Inner Walls Walls In this this case, case, the the influence influenceof ofthe theextremely extremelyhigh highthermal thermalinertia inertia the inner walls is considered, In ofof the inner walls is considered, as as shown in Table 3. The size of the room and the composition of the outer walls remain the same shown in Table 3. The size of the room and the composition of the outer walls remain the same as as in in test Case 1, including identical initial conditions defined by expression difference is test Case 1, including identical initial conditions defined by expression (29).(29). TheThe onlyonly difference is the the increased nominal power of the radiator, since it has to cover additional heat flux losses to the increased nominal power of the radiator, since it has to cover additional heat flux losses to the inner inner walls, even when the steady-state is reached. walls, even when the steady-state is reached. Based on the values for 𝛼 and 𝑆 given in in Table Table3,3, the the additional additional heat heat flux flux in in this this case case is is 𝑖𝑤𝑎 and S 𝑖𝑤𝑎given Based on the values for αiwa iwa ∗ 𝑄 = 500 W and this is exactly the additional required power for a radiator in a heating system ∗ Q𝑖𝑤𝑎 iwa = 500 W and this is exactly the additional required power for a radiator in a heating system regime 80/60/20 °C. outside ◦ C.In regime 80/60/20 Inthis thiscase, case,the thecalculation calculationisis run run as as transient transient with with constant constant prescribed prescribed outside temperature corresponding to the design conditions, 𝑡 = −16 ℃. ◦ 𝑜𝑢𝑡= −16 C. temperature corresponding to the design conditions, tout The results of the calculation proved that the expected values are when obtained when the The results of the calculation proved that the expected values are obtained the steady-state steady-state condition is reached (Figure 8). In Figure 8a, the outside, room, and inner wall condition is reached (Figure 8). In Figure 8a, the outside, room, and inner wall temperatures are given temperatures are given as a function of time (x axis), while in Figure 8b, the components of the total as a function of time (x axis), while in Figure 8b, the components of the total heat flux losses (grey data heat flux (grey data thebar) radiator flux (white data bar) areingiven. Oneeven can note thatthe in bars) andlosses the radiator flux bars) (whiteand data are given. One can note that this case, though this case, even though the temperature difference between the room and inner wall is 1 °C, the ◦ temperature difference between the room and inner wall is 1 C, the contribution of the inner wall contribution of the inner heatheat fluxflux/losses. is more thanThe 23% in total heat flux/losses. The calculation heat flux is more than 23%wall in total calculation results confirmed the expected results and confirmed the results the case is expected validated.results and the case is validated.

(a)

(b)

Figure 8.8.(a)(a) Outside, room, and wall innertemperature; wall temperature; (b) Ventilation, wall, Figure Outside, room, and inner (b) Ventilation, transmissiontransmission wall, transmission transmission windows, internal total losses data bars), radiator flux (white data windows, internal wall flux, total wall lossesflux, (grey data bars),(grey and radiator fluxand (white data bar). bar).

3.3. Validation Case 3—No Thermal Inertia of Inner and Outer Walls This case is designed to validate the use of coupled equations between a multi-layer outer wall and the room assembly at one side and the radiator equation at the other side. For arbitrary variation


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3.3. Validation Case 3—No Thermal Inertia of Inner and Outer Walls This case is designed to validate the use of coupled equations between a multi-layer outer wall and the room assembly at one side and the radiator equation at the other side. For arbitrary variation of the outside temperature, the supply water temperature is determined according to the heat curve parameters and set as a boundary condition for radiator equation. In this case, the size of the room remains the same, while the non-existing thermal inertia of the outer walls is defined by setting the specific heat and density to zero for all four layers of the outer wall. Inner wall heat transfer is neglected by setting Siwa = 0 (see Table 3), and radiator thermal inertia is also neglected by assuming near zero mass of the radiator body and near zero water content in the radiator. As a result of all assumptions, the heat losses for the prescribed room temperature of 20 ◦ C can be calculated by using Equation (30) as: Qloss (τ ) =

n ρcV Uwall Swall + Uwin Swin + h 3600

[20 − tout (τ )]

(30)

where Qloss (τ ) is the heat loss as a function of the time τ and tout (τ ) is the outside temperature variation given as function of time. Buildings 2018, 8, x FOR PEER REVIEW 16 of 18 One can see that the influences of the transient terms are completely neglected by the derivation temperature of 𝑡𝑟𝑜𝑜𝑚 = 20 ℃results for any 𝜏. calculations Also, the radiator heat the flux shouldroom be equal to the of expression (30). The(𝜏) expected of the should keep constant temperature 𝑄𝑙𝑜𝑠𝑠 (𝜏)(τfor of troom 20 ◦𝜏. C for any τ. Also, the radiator heat flux should be equal to the Qloss (τ ) for any τ. ) =any 9a shows shows the variation of the outside temperature and the calculated room temperature, Figure 9a which is is maintained maintained at at the the constant constant and and set set value value of of 20 20◦℃. which C. Variation of room heat losses (circles) calculated by using expression (30) are compared to calculated radiator values, presented calculated by using expression (30) are compared to calculated radiator fluxflux values, presented by aby fulla full in line in Figure can be seen thatexpected the expected are achieved andthe that the Temperature line Figure 9b. It9b. canItbe seen that the resultsresults are achieved and that Temperature Solver Solver satisfied satisfied all tests.all tests. The verification verification and validation validation process was was completed completed and the conclusion is that the the proposed proposed implemented methodology methodology provides provides correct correct results. results. and implemented

(a)

(b)

Figure 9. 9. (a) (a) Room Room and and variable variable outside outside temperature; temperature; (b) (b) Comparison Comparison of of radiator radiator flux flux versus versus total total Figure heat losses. losses. heat

4. 4. Conclusions Conclusions A solution algorithm have beenbeen developed for analysis of single A mathematical mathematicalmodel modeland and solution algorithm have developed for analysis of rooms single in intermittently heated buildings, considering transient effects. The methodology introduces the rooms in intermittently heated buildings, considering transient effects. The methodology introduces additional influence of the thermal capacity of inner walls, which is often neglected in calculations. the additional influence of the thermal capacity of inner walls, which is often neglected in The optimal start of the heating system is influenced temperature of the inner walls calculations. The time optimal start time of the heating systembyisthe influenced by the temperature ofand the can affect energy consumption of the overall building. At the moment, there are no simple guidelines inner walls and can affect energy consumption of the overall building. At the moment, there are no for optimal startingfor ofoptimal the system in order to minimize whileconsumption providing thermal simple guidelines starting of the system in energy order toconsumption minimize energy while providing thermal comfort. This paper is one step forward in this direction, which, combined with measurements of pilot buildings and rooms, can support engineers to find the optimal solution. The second major intention of this paper is to demystify transport energy equations and present them in a simple to solve form. This is especially designed for practitioners whose mathematical skills are lacking, regarding the current needs in the Heating, Ventilation and Air-Conditioning


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comfort. This paper is one step forward in this direction, which, combined with measurements of pilot buildings and rooms, can support engineers to find the optimal solution. The second major intention of this paper is to demystify transport energy equations and present them in a simple to solve form. This is especially designed for practitioners whose mathematical skills are lacking, regarding the current needs in the Heating, Ventilation and Air-Conditioning (HVAC) business. The developed methodology was rigorously tested and validated on cases for which analytical solutions exist. The developed methodology is currently being tested versus measurements in real buildings. The preliminary comparisons of calculated and measured values show promising results which will be the subject of a separate publication. Further development and application will cover an extension of the methodology for calculation of the optimal heat flux from radiators to maintain the desired room temperature, as well as including modeling of thermostatic valves mounted on radiators. Author Contributions: Conceptualization, A.T. and R.B.; methodology, A.T.; software, A.T.; validation, R.B.; formal analysis, R.B.; investigation, A.T. and R.B.; resources, A.T. and R.B.; data curation, A.T.; writing—original draft preparation, R.B.; writing—review and editing, A.T.; visualization, R.B.; supervision, A.T. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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